Balun transformer with variable transformation ratio



April 14, 1964 J. r. COLEMAN 3,129,393

BALUN TRANSFORMER WITH VARIABLE TRANSFORMATION RATIO Filed March 50,1961 2 Sheets-Sheet 1 INVEN TOR. JAMFS 7. COLEMAN April 14, 1964 J. 'r.COLEMAN 3,129,393

BALUN TRANSFORMER WITH VARIABLE TRANSFORMATION RATIO Filed March 30,1961 2 Sheets-Sheet 2 INVENTOR. JAMES r ,rggwA/v United States Patent3,129,393 BALUN TRANSFOIHVIER WITH VARIABLE TRANSvFQRh IATION RATIOJames T. Coieman, Mcorestown, NJL, assignor, by mesne assignments, tothe United States of America as represented by the Secretary of the NavyFiled Mar. 30, 1961, Ser. No. 99,646 11 Claims. (Cl. 333-26) The presentinvention relates generally to apparatus for and methods of couplingradio frequency energy from a balanced to an unbalanced circuit, or viceversa, and more particularly to a balun transformer which possesses animpedance transformation capability.

In numerous radio frequency circuit configurations, it is necessary tocouple a balanced radio frequency source which may, for example,comprise a transmitter embodying tubes arranged in a push-pull manner toan antenna via a relatively low impedance unbalanced line, such as acoaxial cable whose outer conductor is at ground potential. Such anenergy transfer cannot be made directly because neither electricalconnection of the balanced line is grounded and direct connection to thecoaxial cable would immediately remove one side of the balanced line,shorting it to ground.

Devices for permitting the above radio frequency energy transfer areavailable in the prior art and have been variously called chokes,bazookas, line balanced converters and baluns.

Besides the need for converting from a balanced or symmetrical conditionto an unbalanced or asymmetrical condition, these circuits usuallyrequire an impedance matching operation to insure maximum efficiency ofthe over-all system. To perform these complementary functions, some ofthe conventional baluns have been designed to incorporate as an integralcomponent thereof a mechanical tuning feature. However, relativelyinvolved and critical adjustments must be made with these devices tocover only a limited frequency range. Also, some of the typical priorart arrangements for carrying out the impedance matching operationintroduce an undesirable reactance component into the electrical systemwhich destroys its unity power factor and prevents substantially perfectresistive transformations.

It is accordingly a primary object of the present invention to provideapparatus for coupling an unbalanced circuit to a balanced circuit, orvice versa, which also permits the impedances of both circuits to bematched for eflicient energy transformation.

Another object of the present invention is to provide a balanced tounbalanced energy transformer circuit which has a variabletransformation ratio for impedance matching purposes.

A still further object of the present invention is to provide a baluntransformer having a variable transformation ratio.

A still further object of the present invention is to provide a variabletransformation balun transformer whose physical dimensions are only asmall fraction of the wave length of its operating frequency.

A still further object of the present invention is to provide a baluntransformer whose transformation ratio can be continuously varied viaelectrical means.

A still further object of the present invention is to provide a baluntransformer which can match both resistive and reactive loads to thesignal generating source.

A still further object of the present invention is to provide a baluntransformation circuit whose (Q) is low enough for achieving fairly widebandwidth operation- Other objects and many of the attendant advantagesof this invention will be readily appreciated as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings wherein:

FIG. 1 is a schematic circuit diagram of one embodiment of the presentinvention wherein changes in the transformation ratio are accomplishedby a pair of variable capacitors;

FIG. 2 is a simplified drawing of part of the system of FIG. 1 showingonly one loop of the balun transformer;

FIG. 3 is an equivalent circuit of FIG. 1 for facilitating anexplanation of the operating behavior of the invention;

FIG. 4 illustrates an alternative embodiment of the present inventionwhich is closely related to the system of FIG. 1;

FIG. 5 is an alternative construction wherein part of the tuning of thetransformer is done by an inductive reactance;

FIG. 6 shows an arrangement which permits the electrical tuningadjustments to be made outside the high energy field area;

FIG. 7 is a schematic view of an alternative embodiment of the inventionwherein mechanical means are employed to change the impedancetransformation ratio; and

FIG. 8 shows a distributed parametric equivalent of FIG. 1 for ultrahighfrequency use.

Referring now to FIG. 1, a balun transformer arrangement constructedaccording to the present invention is seen to consist of a pair ofparallelly disposed, electrically similar coaxial lines 1 and 2 ofunequal lengths. In the particular case selected for illustration, aradio frequency signal derived [from a source balanced with respect toground is coupled to an unbalanced load. Accordingly, terminals 3 and 4,which are directly connected to the left-hand ends of outer conductors 5and 6 of these coaxial cables, serve as the input connections of thetransformer. Terminal 3 is also connected to the left-hand end of innerconductor 7 of coaxial line 1, and this line is terminated at its otherend in a short circuit formed by part of the surface of conducting plate9. This plate, normally called a shorting plate, is grounded at itsmidpoint 10 and establishes an electrical connection to the outerconductor of coaxial line 2. Coaxial line 2 extends through and beyondthis plate and the load, as represented by resistor R 'is connectedbetween the right-hand terminal end of inner conductor 8 and ground. Theright-hand terminal end of outer conductor -6 is likewise grounded inaccordance with the unbalanced mode of operation. It would be pointedout at this time that the electrical length of coaxial line 1, asmeasured from its left-hand end to shorting plate 9, is only a verysmall fraction of a quarter wave length of the operating radiofrequency.

At the input side of the apparatus, a first variable capacitivereactance, provided by condenser C is connected between the innerconductors 7 and 8; while a second variable capacitive reactance,provided by condenser C is connected between the confronting portion ofthe left-hand ends of the outer conductors 5 and 6. It will berecognized, of course, that in the receiving mode of operation resistorR would be replaced with an input circuit and the output taken acrossterminals 3, 4.

In order to understand the principles of operation of the above circuit,reference may now be had to FIG. 2, which shows only the input loop ofthe balun transformer as viewed from terminals 3 and 4. Thus, with abalanced radio frequency source connected across these terminals, acurrent path may be traced from terminal 3 down the outer and innerconductors 5 and 7 of line 1 to and across shorting plate 9 and back toterminal 4 via the outer conductor only of coaxial line 2, disregardingfor the time being the effect of capacitor C The impedance lookingPatented Apr. 14, 1964 3,129,393 1% a into these terminals is that of ashort-circuit transmisand sion line because of the conducting pathestablished by X plate 9. This impedance, as is well known, is (9) Sin34 1 n all "RoztXcf with Z0 the Characteristic impedance of the twoconduc 5 and substituting these last two relationships in Equation 7 torline formed by the outer conductors, We obtam (10)- 2Il' R0 X01 A Y-R02+XC12+ o 'i' C/ with X t free Space Wave length of the fafiio f q y 10The ideal transformer places the admittance Y in parenergy In meters andl h g 0f the 11116 meters, allel with the admittance Y when the circuitis viewed In the Case 10S$ 1eS$ 11116, 11118 Impedance is l from thebalanced terminals 3, 4 and, consequently, these to be a pure inductivereactance. When capacitor C is admittances can b bi as taken intoaccount, this impedance is paralleled with a 15 11 variable capacitivereactance. The admittance across these terminals can be stated as R e 11 (2) Ya4= 'l'jwC'z 34 R2+Xc12 JR2+XC12 tan m 7X62 J O tan I 1 0 Now, ifthe sum of the last three terms in Equation 11 is set to equal zero bytuning the two variable capacitors, Whlch can be rewntten as 1 1 thenonly the real part of the admittance, that is, the conductance G,appears across terminals 3, 4 in the form (3) Y34 Z tan {3Z X ofConsidering once again the behavior of the system of (1 r FIG. 1, theradio frequency current path in the second C1 loopmay be traced from theungrounded side of load reand the load resistor R is now transformedinto sistor R through the inner conductor 8 of line 2, variable R 2 X 2capacitor C the inner and outer conductors 7 and 5, re- 3 L Espectively, of line 1 to shorting plate 9, and then across 0 this platedown alon the outer conductor 6 of line 2 to the left-end terminatingedge thereof, and thence, via RIMQER" Whlch of coursa equal to or greatthan the inner surface of this same outer conductor back to the groundedside of the load resistor. Thus, the sys- .From a Study of the abovemimiemailcal analysis It i behaves as a very tightly Coupled 01164043116ram} 35 will be undertsood that a pure resistive load can be madetransformen It would be pointed out at this time that no to appaar thebalanced i .that the: magmload current flows between the shorting plateto the ture of this resistance can be widely varied by selectinggrounded side of the load resistor because of the potential dlfierentvalues of C1 and that lhmllghout i changes a equality existing betweenthese points brought about by umty pgwer factor can be mamtamed yslrmiar adjust" the balanced Configuration ments to C By the same token,a low input impedance The balun transformer of FIG. 1 may be representedClrcult Connected to h unbaiancfid q of h balun by the equivalentcircuit of FIG 3 The admittance: transformer can have its magnitudecontinually increased Y appsars across the primary Winding of an idealtrans and made to appear in balanced form at the other side of former 15and the series combination of capacitor C and p the -q the loadresistance R0 appears across thg secondary wind 40 A slight variation inthe circuit of FIG. 1 is shown in ing. In the following mathematicalpresentation, the 00- F Here the variable capacitive reactance axialline 2 will be considered as being terminated in its nected acfross the9 Conductors 5 i 6 takes the form characteristic impedance R and theelectrical length of of 3 genes combmatlon of two vanable Condensers C3the short-circuited transmission line as being only a and Each of thesecqmponents course has half Small fraction of M4. thereactance of C ortwice the capacitance because of The impedance of the series wmbinationof c1 and ilIlC1l' SIl S relationship and the midpoint of the series iscombination is grounded. This configuration gives a better circuitbalance than that realized with the arrangement o' C1 of FIG. 1. whichcan be rewritten in Polar form as Another variation of the originalcircuit is schematically depicted inFIG. 5 and, in this particularmodification, the i Z =V 5 X 0 :3 variable capacitive reactance C isreplaced with a variable The admittance of this combination is thereciprocal of Inductive madam? The f wmbmanon of capacl ,[116 aboveimpedance that is, tors C and C is still maintained; however, they aremade large enough to make Y capacitive for this case. 1 In FIG. 6 thereis schematically illustrated a modification similar to that of FIG. 1wherein the coaxial line 1 and substituting for Z in this lastrelationship we obtain does not tflminam in a ShOItcircuit but extendsthrough plate 9 and is terminated in a variable capacitive reactance,(6) y l. l 6 C In order to avoid the necessity of any mechanical V o Ctuning, variable capacitive reactance C is again connected which can beexpressed as across the outer conductors of the coaxial lines at thebalanced ends thereof. Hence, in this configuration, too, 05 6 i 6 allthe tuning is done by electrical means, namely, con- W {W densers C andC The purpose of placing the C at 1 1 the extremity of coaxial line 1 isto permit the tuning to but be made outside of the high radio frequencyfield area,

C 9 R In FIG. 7 there is illustrated a balun transformer circuit (8) 0Swherein the tuning is accomplished by moving shorting +XC1 bar 9 todifferent positions along the outer conductors 5 and 6 of coaxial lines12 and 13. In this particular configuration, the mechanical tuningfeature eliminates the need of variable condenser C which otherwisewould be required across outer conductors 5 and 6. Likewise, thevariable capacitive reactance provided by condenser C is hererepresented by a transmission line equivalent in the form of theopen-circuited coaxial line 12 of length In regards to this lastsubstitution, it is well known that such an open-circuited transmissionline presents a capacitive reactance if it is less than a quarter wavelength long at the operating frequency. It is this characteristic whichprovides the system with the capacitive reactance previously supplied bycondenser C in the second loop of the transformer as depicted in FIG. 3.It would be observed that with 1 less than a quarter wave length acapacitive reactance appears between terminals 16 and 17 of theapparatus and that the -jX term of expression (4) is now 'Z cot ,Blwhere Z is the characteristic impedance of coaxial line 12. Theadmittance across terminals 3, 4 can now be rewritten as 14) Y JZOZ tan61 where Z is now the characteristic impedance of coaxial lines 12 and13 between outer conductors. The matching performance of the transformernow becomes 15 R.,, L d R0 and for unity power factor Z01 (30b 6Z 1 Itwill be appreciated that for proper operation of the balun transformer land 1 are adjusted to satisfy Equations 15 and 16.

In designing the apparatus of FIG. 7, the values of R' and R would beknown and Z selected so as to give convenient values of l over theoperating frequency range. Also, Z would be selected for convenientvalues of 1 over this same frequency range; hence, the preciserelationship of 1 and would be defined. The above expressions from aphysical viewpoint are good to the case where Z is about equal to thespace in between centers of coaxial lines 12 and 13. This relationshipwill give the shortest dimension for It would be noted that with theapparatus of FIG. 1 can be much smaller than the aforementioned length,attaining values of about M30. The reason for this being that condenserC makes coaxial lines 1 and 2 of FIG. 1 appear electrically longer.

FIG. 8 shows a balun transformer designed for ultrahigh frequencyoperation employing only transmission line components. This apparatus isa distributed parametric equivalent of the transformer of FIG. 1 and,like the arrangement of FIG. 7, it employs a movable shorting plate 9 totune the transformer. The required inductive reactance across terminals21 and 22, the left-hand ends of the outer conductors of coaxial lines23 and 24, is achieved by shorting bar 9 which maintains the length at avalue less than 4, as is well known. Likewise, the electrical length ofcoaxial line 23, as determined by the axial position of internalshorting bar 25 and as represented by dimension Z is kept between M4 andM2 in order to present capacitive reactance between terminals and 21.The last requirement, of course, comes about from the well-knoum factthat a shorted transmission line between M4 and M2 acts as a purecapacitive reactance. One of the noteworthy advantages of thearrangement of FIG. 8 is that the variable matching property can be usedat very high frequencies, such as 600 mc./s. where M 4 is 4.92 inches.Another characteristic which recommends the configuration of FIG. 8 isfound in the fact that the short-circuited termination employed withcoaxial line 23 is much easier to construct than the open-circuitedtermination of coaxial line 12 in FIG. 7.

At this point, it would be mentioned that the apparatus of FIGS. 1, 4, 5and 6 may be preferred over the other structures because of theavailability of vacuum variable capacitors whose parameters can bevaried over considerably wide ranges. For example, units made by theJennings Radio, of Palo Alto, California, have a capacitance range from25 to 450 mfd. By resorting to: such lumped variable electricalcomponents, the physical size of the balun can be greatly reduced,reaching dimensions which are as short as 30. And for example, in thearrangement of FIG. 1 can be made at least as small as the separationbetween coaxial lines 5 and 6 as measured between line centers.

If desired, the inner conductor 7 of coaxial line 1 in FIGS. 1, 4 and 5can be eliminated without altering the performance of the baluntransformer. Capacitor C would then be connected between the left-handend of inner conductor 8 of coaxial line 2 and the outer conductor 5 ofcoaxial line 1.

Obviously many modifications and variations of the? present inventionare possible in the light of the above teachings. It is therefore to beunderstood that within the scope of the appended claims the inventionmay be practiced otherwise than as specifically described.

What is claimed is:

1. A balun transformer comprising, in combination, a pair of coaxiallines of unequal length, a conducting plate connecting the outerconductors of said coaxial line together at a given distance from oneend of each coaxial line, said conducting plate being grounded andshortcircuiting the inner and outer conductors of the shorter coaxialline at the other end thereof, means connecting the inner and outerconductors of the shorter coaxial line together at said one end thereof,first and second variable capacitors, said first capacitor beingconnected between the inner conductors of said coaxial lines at said oneend thereof, said second capacitor being connected between the outerconductors of said coaxial lines at said one end thereof, means forcoupling a radio frequency signal across the outer conductors of saidcoaxial lines at said one end thereof, the output of said transformerbeing taken across the inner and outer conductors of the longer coaxialline at the other end thereof.

2. A balun transformer having a variable transformation ratiocomprising, in combination, first and second coaxial lines, a firstvariable capacitor connected across the inner conductors of said coaxiallines at one end thereof, a second variable capacitor connected acrossthe outer conductors of said coaxial lines at said one end thereof,means for connecting the inner and outer conductors of the first coaxialline together at said one end thereof, means for coupling a radiofrequency signal across said second variable capacitor, a conductingplate connected between the outer conductors of said coaxial lines at adistance from said one end thereof, said distance 1 being less than onequarter of the wave length of said radio frequency signal, saidconducting plate being grounded and short-circuiting the inner and outerconductors of said first coaxial line at the other end thereof, meansfor grounding the outer conductor of the second coaxial line at theother end thereof, and means for taking an output signal between theinner and outer conductors of said second coaxial line at the other endthereof.

3. A balun transformer comprising, in combination, a pair of metallictubular members of unequal length, the longer member of said pair havinga concentric conductor disposed therein, a first variable capacitorconnected between one end of the shorter tubular member and one end ofsaid conductor, a second variable capacitor connected between said oneend of the shorter tubular conductor and that end of the longer tubularconductor which is adjacent said one end of said conductor, a conductingplate joining said first and second tubular members together, said plateand said tubular members forming a short-circuited transmission line,means for feeding a radio frequency signal between said one end of saidshorter tubular member and that end of said larger tubular conductorwhich is adjacent said one end of said conductor, means for groundingsaid conducting plate, means for taking an output signal from across theother end of said concentric conductor and the other end of said shortertubular member.

4. A balun transformer comprising, in combination, first and secondcoaxial lines, means for connecting the inner conductors of said coaxiallines together at first ends thereof, means for coupling a radiofrequency signal across the outer conductors of said coaxial lines atsaid first ends, a conducting plate connecting the outer conductors ofsaid coaxial lines together, said plate being movable along said outerconductors whereby said plate and said outer conductors form ashort-circuited transmission line of length means for grounding theouter conductor of the first coaxial line at the second end thereof, aload connected between the inner and outer conductors of said firstcoaxial line at the second end thereof, said second coaxial lineterminating in an open circuit at the second end thereof and having alength which is less than a quarter Wave length of said radio frequency.

5. A balun transformer comprising, in combination, first and secondcoaxial lines of unequal length, a first variable capacitor connectedbetween a first end of the outer conductor of the shorter coaxial lineand a first end of the outer conductor of the lon er coaxial line, meansfor connecting the inner conductors of said coaxial line together atsaid first ends, means for coupling a radio frequency signal across theouter conductors of said coaxial lines at said first ends thereof, aconducting plate connecting the outer conductors of said coaxial linestogether at a distance 1 from said first ends thereof, said distance 1being less than a quarter wave length of the radio frequency energyconnected across the outer conductors at said first ends thereof, asecond variable capacitor connected between the inner and outerconductors of the shorter coaxial line at the second end thereof, and aload connected between the inner and outer conductors of the longercoaxial line at the second end thereof, and means for grounding theouter conductor of the longer coaxial line at said second end thereof.

6. A balun transformer comprising, in combination, first and secondcoaxial lines of unequal length, means for coupling a radio frequencysignal across corresponding first ends of the outer conductors of saidcoaxial lines, means for connecting the inner conductors of said coaxiallines together at said first ends, a conducting plate connecting theouter conductors of said coaxial lines together, said conducting platebeing movable along said outer conductors thereof to form in combinationwith said outer conductors a short-circuited transmission line ofvariable length, means for short-circuiting the inner and outerconductors of the longer coaxial line at variable distances from thesecond end of said coaxial line, a load connected between the inner andouter conductors of said shorter coaxial line at the second end thereof,the length of the short-circuited transmission line being less than onequarter wave length of said radio frequency energy and the distance fromthe first end of the longer coaxial line and the location of the shortcircuit being between one quarter and a half wave length of said radiofrequency energy.

7. A balun transformer comprising, in combination, first and secondcoaxial lines, means for connecting the inner conductors of said coaxiallines together at one end thereof, means for coupling a radio frequencysignal across the outer conductors of said coaxial lines at said one endthereof, a conducting plate connecting the outer conductors of saidcoaxial lines together, said plate being movable along said outerconductors whereby said outer conductors and said plate form ashort-circuited trans mission line of length 1 said length 1 being lessthan a quarter Wave length of said radio frequency signal, means forgrounding the outer conductor of the first coaxial line at the other endthereof, a load connected between the inner and outer conductors of saidfirst coaxial line at the other end thereof, said second coaxial linebeing open at its other end and having a length l which is less than aquarter wave length of said radio frequency signal.

8. A balun transformer having a variable transformation ratiocomprising, in combination, a first and second hollow conductor, aconcentric conductor disposed Within said first hollow conductor, afirst variable capacitor, said first variable capacitor being connectedbetween a first end of said concentric conductor and a first end of saidsecond hollow conductor, means for coupling a radio frequency signalbetween a first end of said first hollow conductor and said first end ofsaid second hollow conductor, said first end of said concentricconductor and said first end of said first hollow conductor beingcoadjacent, second and third variable capacitors, said second capacitorbeing connected between said first end of said first hollow conductorand ground potential, said third variable capacitor being connectedbetween said first end of said second hollow conductor and groundpotential, a conducting plate bridging said first and second hollowconductors at a distance from the first ends of said hollow conductors,said distance 1 being less than a quarter wave length of the frequencyof said radio frequency signal, means for connecting the second end ofsaid first hollow conductor to ground potential and a load connectedbetween the second end of said concentric conductor and groundpotential.

9. A balun transformer having a variable transformation ratiocomprising, in combination, a first and second hollow conductor, aconcentric conductor disposed within said first hollow conductor, avariable inductance, said variable inductance being connected between afirst end of said concentric conductor and a first end of said secondhollow conductor, means for coupling a radio frequency signal between afirst end of said first hollow conductor and said first end of saidsecond hollow conductor, said first end of said concentric conductor andsaid first end of said first hollow conductor being coadjacent, firstand second variable capacitors, said first capacitor being connectedbetween said first end of said first hollow conductor and groundpotential, said second variable capacitor being connected between saidfirst end of said second hollow conductor and ground potential, aconducting plate bridging said first and second hollow conductors at adistance 1 from the first ends of said hollow conductors, said distancel being less than a quarter wave length of the frequency of said radiofrequency signal, means for connecting the second end of said firsthollow conductor to ground potential and a load connected between thesecond end of said concentric conductor and ground potential.

10. A balun transformer comprising, in combination, first and secondtubular members of unequal length, said first tubular member beingshorter than said second tubular member, concentric conductors disposedwithin said first and second tubular members, means interconnecting afirst end of said first tubular member and a first end of the concentricconductor disposed therein, a variable capacitor connected between afirst end of said second tubular member and said first end of said firsttubular member, a second variable capacitor connected between said firstend of said first tubular member and a first end of the concentricconductor disposed Within said second tubular member, means for couplinga radio frequency signal across said first end of said first tubularmember and said first end of said second tubular member, a conductingplate connecting said first and second tubular members together at adistacne I from said first ends of said tubular members, the secc'ildend of the concentric conductor disposed within said first tubularmember connected to said conducting plate, said conducting plate beinggrounded at its midpoint and closing the second end of said firsttubular member, means for grounding the second end of said secondtubular member, and means for connecting a load between said second endof said second tubular member and the second end of the concentricconductor disposed therein.

11. An arrangement as defined in claim 10 wherein the distance 1 is lessthan a quarter Wave length of the radio frequency energy connectedacross the first ends of said tubular members.

No references cited.

1. A BALUN TRANSFORMER COMPRISING IN COMBINATION, A PAIR OF COAXIALLINES OF UNEQUAL LENGTH, A CONDUCTING PLATE CONNECTING THE OUTERCONDUCTORS OF SAID COAXIAL LINE TOGETHER AT A GIVEN DISTANCE FROM ONEEND OF EACH COAXIAL LINE, SAID CONDUCTING PLATE BEING GROUNDED ANDSHORTCIRCUITING THE INNER AND OUTER CONDUCTORS OF THE SHORTER COAXIALLINE AT THE OTHER END THEREOF, MEANS CONNECTING THE INNER AND OUTERCONDUCTORS OF THE SHORTER COAXIAL LINE TOGETHER AT SAID ONE END THEREOF,FIRST AND SECOND VARIABLE CAPACITORS, SAID FIRST CAPACITOR BEINGCONNECTED BETWEEN THE INNER CONDUCTORS OF SAID COAXIAL LINES AT SAID ONEEND THEREOF, SAID SECOND CAPACITOR BEING CONNECTED BETWEEN THE OUTERCONDUCTORS OF SAID COAXIAL LINES AT SAID ONE END THEREOF, MEANS FORCOUPLING A RADIO FREQUENCY SIGNAL ACROSS THE OUTER CONDUCTORS OF SAIDCOAXIAL LINES AT SAID ONE END THEREOF, THE OUTPUT OF SAID TRANSFORMERBEING TAKEN ACROSS THE INNER AND OUTER CONDUCTORS OF THE LONGER COAXIALLINE AT THE OTHER END THEREOF.